The unit circle is a good mental tool for this.
cos(x) = -1/2 --> x = 2pi/3
Now, you know that in the interval 0 to 2pi the cosine-function is negative in the second and third quadrants. That is,
cos(2pi/3) = cos(-2pi/3)
The inequality thus holds true for the interval
-2pi/3 < x < 2pi/3
just like you think. Draw a unit circle and plot the values above.
Not sure how used you are to working with trigs, give a shout if you need more clarification.